Spectral transitions of the entanglement Hamiltonian in monitored free fermions

We numerically investigate measurement-induced phase transitions in monitored free fermions through the spectral and eigenstate properties of the entanglement Hamiltonian. By analyzing entanglement scaling, we identify three nontrivial fixed points, for which the entanglement follows exact analytical scaling forms: chaotic unitary dynamics at infinitesimal monitoring, characterized by a Gaussian Page law; a Fermi-liquid fixed point at moderate monitoring, defining a metallic phase with logarithmic entanglement growth and emergent space-time invariance; and a quantum Lifshitz fixed point marking the measurement-induced phase transition into a localized area-law phase. Adopting a random-matrix perspective on the entanglement Hamiltonian, we show that short-range spectral correlations, such as the adjacent gap ratio $\ensuremath{\langle}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{r}\ensuremath{\rangle}$ and the Kullback-Leibler divergence ${\text{KL}}_{1}$, sharply detect an ergodic to nonergodic phase transition at the quantum Lifshitz fixed point, and yield precise estimates of the critical point and correlation length exponent. Long-range probes, including the spectral form factor and the associated Thouless time, corroborate this picture, while the variant ${\text{KL}}_{2}$ uncovers signatures of a possible nonergodic extended (multifractal) regime at intermediate monitoring strengths. Together, these results establish the entanglement Hamiltonian as a powerful framework for diagnosing metallic, localized, and multifractal regimes in monitored quantum dynamics, and highlight unexpected scaling structures, Fermi liquid, and Lifshitz criticality, that lie beyond current field-theoretic approaches.